Slate piece on martingales, expected value, and the bailout

Here’s how to make money flipping a coin. Bet 100 bucks on heads. If you win, you walk away $100 richer. If you lose, no problem; on the next flip, bet $200 on heads, and if you win this time, take your $100 profit and quit. If you lose, you’re down $300 on the day; so you double down again and bet $400. The coin can’t come up tails forever! Eventually, you’ve got to win your $100 back.

Or not. If you want to see how well this strategy works in practice (answer: not very) try a few runs of the martingale applet at UIUC.

My colleague Timo Seppalainen explained to me a nice way of seeing of the long-term failure of the martingale. Let X_j be the length of the jth run of tails. Then X_j is 0 with probability 1/2, 1 with probability 1/4, 2 with probability 1/8, and so on. The chance that X_j >= n is 1/2^n. In particular, the probability that X_j is at least (log_2 j + 1) is about 1/2j.

But the amount of money you lose on a run of n tails is about 2^n, while the amount of money you’ve won prior to the start of the jth run is about j. In particular, if X_j > (log_2 j + 1) then you’re at least j dollars down after the jth run of tails. Since the sum of 1/2j as j goes to infinity diverges, you almost always have infinitely many occurences of X_j > (log_2 j + 1); as I learned from Timo, this follows from the second Borel-Cantelli Lemma.

So there are infinitely many j such that, after the jth run of tails, you’re at least j dollars down. Even if you start with a million dollars, that means you’re eventually going broke.

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3 thoughts on “Slate piece on martingales, expected value, and the bailout”

Actually, this type of martingale game is one of the things that the naturalist Buffon (that I’ve been posting about recently) had terrible problems with. He clearly felt there was something wrong with it, but his explanation is mathematically rather weirder. (I will describe it in a post fairly soon).

Put more simply: there is a non-zero probability that you’ll have a run of n tails, which will bankrupt you if you have a stake (original or enhanced) less than 2^n, for any n. So unless you have an infinite amount of money to risk (in which case why play?), don’t bother.

I think the simple explanation is too simple: just because a financial proposition has a positive chance of bankrupting you doesn’t in itself make it undesirable proposition. Part of the point here is that the risk of bankruptcy is not sufficiently offset by the positive outcomes: you’re trying to turn a situation with mathematical expectation zero into a situation with positive expectation via some betting pattern, and that’s impossible (although the impossibility is intriguingly nonobvious to prove or even to state in a precise and sufficiently general form).

By way of comparison, suppose that we flip a coin once. If it comes up heads, you give me a million dollars. If it comes up tails we flip it again. If the second flip comes up heads we both walk away unscathed, but if the second flip comes up tails then I give you all my money.

Clearly the chance of bankruptcy is an extremely non-neglible 25%. But, so long as my net worth is less than $2 million, the expectation of this transaction is positive. One interesting thing here is that whether this proposition is desirable to me depends upon more than just my net worth — it depends upon how risk averse I am and also on more practical matters like what would happen to me if I lost all my money.

In fact, in my particular situation, I think I would probably accept such a proposition, since my current net worth is substantially less than $2 million but I have a steady job, no dependents, people who would put me up while I waited for my next paycheck to come in, etc. If instead of $1 million the winnings were $10 million, I would accept the proposition very enthusiastically. I would think you could get almost anyone to accept this proposition by making the potential winnings large enough and the probability of bankruptcy small enough, but still positive.